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Properties

Label 2900.2.f.d
Level $2900$
Weight $2$
Character orbit 2900.f
Analytic conductor $23.157$
Analytic rank $0$
Dimension $8$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2900,2,Mod(1449,2900)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2900.1449"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2900, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2900 = 2^{2} \cdot 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2900.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,0,0,0,0,24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(23.1566165862\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.205520896.4
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 12x^{6} - 12x^{5} - 8x^{4} + 12x^{3} + 12x^{2} + 4x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 580)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{7} q^{3} + (\beta_{2} - \beta_1) q^{7} + ( - 2 \beta_{3} + 3) q^{9} + \beta_{5} q^{11} - 4 \beta_{2} q^{13} + (2 \beta_{7} - \beta_{4}) q^{17} + ( - \beta_{6} + 2 \beta_{5}) q^{19} + ( - \beta_{6} - \beta_{5}) q^{21}+ \cdots + ( - 2 \beta_{6} + \beta_{5}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 24 q^{9} - 8 q^{29} + 8 q^{49} - 80 q^{51} - 16 q^{59} + 48 q^{71} - 8 q^{81} + 64 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 12x^{6} - 12x^{5} - 8x^{4} + 12x^{3} + 12x^{2} + 4x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -12\nu^{7} + 28\nu^{6} - 52\nu^{5} - 158\nu^{4} + 524\nu^{3} - 268\nu^{2} - 364\nu - 88 ) / 51 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 32\nu^{7} - 137\nu^{6} + 422\nu^{5} - 508\nu^{4} - 94\nu^{3} + 335\nu^{2} + 336\nu + 76 ) / 51 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -2\nu^{7} + 9\nu^{6} - 28\nu^{5} + 36\nu^{4} + 4\nu^{3} - 33\nu^{2} - 10\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -35\nu^{7} + 161\nu^{6} - 503\nu^{5} + 664\nu^{4} + 55\nu^{3} - 623\nu^{2} - 189\nu + 140 ) / 51 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3\nu^{7} - 11\nu^{6} + 31\nu^{5} - 20\nu^{4} - 49\nu^{3} + 43\nu^{2} + 51\nu + 12 ) / 3 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3\nu^{7} - 13\nu^{6} + 41\nu^{5} - 52\nu^{4} + \nu^{3} + 29\nu^{2} + 27\nu + 6 ) / 3 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -61\nu^{7} + 267\nu^{6} - 831\nu^{5} + 1030\nu^{4} + 159\nu^{3} - 909\nu^{2} - 275\nu - 28 ) / 51 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} - 2\beta_{2} + \beta _1 + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} - 3\beta_{3} - 3\beta_{2} + \beta _1 - 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 5\beta_{7} + \beta_{6} - 5\beta_{5} + 9\beta_{4} - 18\beta_{3} + 4\beta_{2} - 7\beta _1 - 22 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -4\beta_{6} - 16\beta_{5} + 24\beta_{2} - 21\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -37\beta_{7} - 7\beta_{6} - 37\beta_{5} - 81\beta_{4} + 150\beta_{3} + 52\beta_{2} - 49\beta _1 + 202 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -43\beta_{7} + 43\beta_{6} + 91\beta_{5} - 91\beta_{4} + 171\beta_{3} - 171\beta_{2} + 113\beta _1 + 226 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 103\beta_{7} + 305\beta_{6} + 713\beta_{5} + 305\beta_{4} - 504\beta_{3} - 1286\beta_{2} + 895\beta _1 - 782 ) / 4 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2900\mathbb{Z}\right)^\times\).

\(n\) \(901\) \(1277\) \(1451\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1449.1
1.55050 − 0.642239i
1.55050 + 0.642239i
−0.129640 − 0.312979i
−0.129640 + 0.312979i
1.12964 + 2.72719i
1.12964 − 2.72719i
−0.550501 + 0.228025i
−0.550501 − 0.228025i
0 −2.97127 0 0 0 − 0.585786i 0 5.82843 0
1449.2 0 −2.97127 0 0 0 0.585786i 0 5.82843 0
1449.3 0 −1.78089 0 0 0 − 3.41421i 0 0.171573 0
1449.4 0 −1.78089 0 0 0 3.41421i 0 0.171573 0
1449.5 0 1.78089 0 0 0 − 3.41421i 0 0.171573 0
1449.6 0 1.78089 0 0 0 3.41421i 0 0.171573 0
1449.7 0 2.97127 0 0 0 − 0.585786i 0 5.82843 0
1449.8 0 2.97127 0 0 0 0.585786i 0 5.82843 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1449.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
29.b even 2 1 inner
145.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2900.2.f.d 8
5.b even 2 1 inner 2900.2.f.d 8
5.c odd 4 1 580.2.d.b ✓ 4
5.c odd 4 1 2900.2.d.c 4
15.e even 4 1 5220.2.l.g 4
20.e even 4 1 2320.2.g.d 4
29.b even 2 1 inner 2900.2.f.d 8
145.d even 2 1 inner 2900.2.f.d 8
145.h odd 4 1 580.2.d.b ✓ 4
145.h odd 4 1 2900.2.d.c 4
435.p even 4 1 5220.2.l.g 4
580.o even 4 1 2320.2.g.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
580.2.d.b ✓ 4 5.c odd 4 1
580.2.d.b ✓ 4 145.h odd 4 1
2320.2.g.d 4 20.e even 4 1
2320.2.g.d 4 580.o even 4 1
2900.2.d.c 4 5.c odd 4 1
2900.2.d.c 4 145.h odd 4 1
2900.2.f.d 8 1.a even 1 1 trivial
2900.2.f.d 8 5.b even 2 1 inner
2900.2.f.d 8 29.b even 2 1 inner
2900.2.f.d 8 145.d even 2 1 inner
5220.2.l.g 4 15.e even 4 1
5220.2.l.g 4 435.p even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 12T_{3}^{2} + 28 \) acting on \(S_{2}^{\mathrm{new}}(2900, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} - 12 T^{2} + 28)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} + 12 T^{2} + 4)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 20 T^{2} + 28)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 32)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} - 52 T^{2} + 28)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 76 T^{2} + 1372)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 36 T^{2} + 196)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 4 T^{3} + \cdots + 841)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 12 T^{2} + 28)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 20 T^{2} + 28)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 40 T^{2} + 112)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 12 T^{2} + 28)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 12 T^{2} + 28)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 48 T^{2} + 64)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 4 T - 28)^{4} \) Copy content Toggle raw display
$61$ \( (T^{4} + 40 T^{2} + 112)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 132 T^{2} + 3844)^{2} \) Copy content Toggle raw display
$71$ \( (T - 6)^{8} \) Copy content Toggle raw display
$73$ \( (T^{4} - 332 T^{2} + 26908)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 244 T^{2} + 14812)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 300 T^{2} + 2500)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 384 T^{2} + 28672)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 332 T^{2} + 26908)^{2} \) Copy content Toggle raw display
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